Conditions which imply metrizability in some generalized metric spaces (Q2701853)

A Hausdorff space \((X,\tau)\) is called a strongly-quasi Nagata space if it has a function \(g:\omega\times X\to\tau\) with the following properties:NEWLINENEWLINENEWLINE(a) \(x\in g(n,x)\) for each \(x\in X\) and for each \(n<\omega\);NEWLINENEWLINENEWLINE(b) \(g(n+1,x) \subset g(n,x)\) for each \(x\in X\) and for each \(n<\omega\);NEWLINENEWLINENEWLINE(c) if for each \(n<\omega\), \(y_n\in g(n,x_n)\) and the sequence \((y_n)_{n <\omega}\) converges to \(x\) in \(X\), then \(x\) is a cluster point of the sequence \((x_n)_{n <\omega}\).NEWLINENEWLINENEWLINEIn [Quest. Answers Gen. Topology 5, 281-291 (1987; Zbl 0643.54035)] it was shown by \textit{Z. Gao} that every regular \(k\)-semistratifiable Hausdorff space is a strongly-quasi Nagata space. This paper contains the following additional results concerning strongly-quasi Nagata spaces: (1) Every subspace of a strongly-quasi Nagata space is a strongly-quasi Nagata space. (2) Every countable product of strongly-quasi Nagata spaces is a strongly-quasi Nagata space. (3) Closed images of regular strongly-quasi Nagata spaces are strongly-quasi Nagata spaces. (4) Every strongly-quasi Nagata space is a \(\sigma\)-space. (5) A Hausdorff space is Nagata space if and only if it is a first countable strongly-quasi Nagata space. (6) A Hausdorff space is metrizable if and only if it is a strongly-quasi Nagata space and a quasi-\(\gamma\)-space. Using (6) the following theorem is proved: (7) A quasi-\(\gamma\)-space is metrizable if and only if it is a pseudo \(wN\)-space or quasi-Nagata-space with quasi-\(S_2\)-diagonal. This is an affirmative answer to a question of \textit{H. W. Martin} [Proc. Am. Math. Soc. 57, 332-336 (1976; Zbl 0334.54015)]. (8) For a \(q\)-space with quasi-\(G_\delta^*\)-diagonal the properties of being a Nagata space, a strongly-quasi Nagata space, or a quasi-Nagata-space are all equivalent.